On Statistical Testing of Random Numbers Generators

  • F. El Haje
  • Y. Golubev
  • P. -Y. Liardet
  • Y. Teglia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4116)


Maurer’s test is nowadays a basic statistical tool for testing physical random number generators in cryptographic applications. Based on a statistical analysis of this test we propose simple and effective methods for its improvement. These methods are related to the m – spacing technique common in goodness-of-fit problems and the L – leave out method used for a noise reduction in the final Maurer test statistic. We also show that the spacing distribution test represents a serious competitor for Maurer’s test in the case when the random number generator is governed by a Markov chain with a long memory.


Error Probability Random Number Generator Block Length Markov Chain Model Uniformity Testing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cressie, N.: On the logarithms of high-order spacings. Biometrika 63(2), 343–355 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Coron, J.-S., Naccache, D.: An Accurate Evaluation of Maurer’s Universal Test. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 57–71. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Coron, J.-S.: On the Security of Random Sources. In: Imai, H., Zheng, Y. (eds.) PKC 1999. LNCS, vol. 1560, p. 29. Springer, Heidelberg (1999), CrossRefGoogle Scholar
  4. 4.
    Del Pino, G.: On the asymptotic distribution of k-spacings with applications to goodness-of-fit tests. Ann. Statist. 8, 1058–1065 (1979)CrossRefGoogle Scholar
  5. 5.
    Goubin, L., Patarin, J.: DES and Differential Power Analysis the ”Duplication” Method. In: Proceedings of CHESS 1999. LNCS. Springer, Heidelberg (1999)Google Scholar
  6. 6.
    Finner, H., Rotters, M.: Multiple hypothesis testing and expected number of type 1 errors. Ann. Statist. 30(1), 220–238 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hall, P.: limit theorems for sums of general functions of m-spacings. Math. Proc. Cambridge Phil. Society 96, 517–532Google Scholar
  8. 8.
    Maurer, U.: A universal statistical test for random bit generators. J. Cryptology 5(2), 89–105 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    National Institute of Standards and Technology, Security Requirements for Cryptographic Modules, FIPS 140-1 (January 1994),
  10. 10.
    National Institute of Standards and Technology, Security Requirements for Cryptographic Modules, FIPS 140-2 (May 2001),
  11. 11.
    Functionality Classes and Evaluation Methodology for Deterministic Random Number Generators, Version 2.0 (December 1999),
  12. 12.
    Functionality Classes and Evaluation Methodology for True (physical) Random Number Generators, Version 3.1 (September 2001),
  13. 13.
    Oswald, E., Preneel, B.: A Survey on Passive Side Channel Attacks and their Counter Measures for the Nessie Public-Key Cryptosystems,
  14. 14.
    Pyke, R.: Spacings. J.R. Statist. Soc. B 27, 395–436 (1965)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Shilling, M.: Goodness of fit in ℝn based on weighted empirical distributions of certain neighbor statistics. Ann. Statist. 11, 1–12 (1983)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Weggenkitl, S.: Entropy estimators and Serial Tests for Ergodic Chains. IEEE Trans. Inform. Theory 47(6), 2480–2489 (2001)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Weiss, L.: Asymptotic of certain tests of fit based on sample spacings. Ann. Math. Statist. 28, 783–786 (1957)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • F. El Haje
    • 1
    • 3
  • Y. Golubev
    • 1
  • P. -Y. Liardet
    • 2
  • Y. Teglia
    • 2
  1. 1.Université de Provence (Aix-Marseille 1), CMIMarseilleFrance
  2. 2.STMicroelectronics, ZI RoussetRoussetFrance
  3. 3.Centre Microélectronique de Provence, Georges CharpakGardanneFrance

Personalised recommendations